Resum:

When collecting survey data for a specific study it is usual to have some background information, in the form for example, of socio-demographic variables. In our context, these variables may be useful in identifying potential sources of heterogeneity. Resolving the heterogeneity may mean to perform distinct analyses based on the main variables for distinct and homogeneous segments of the data, defined in terms of the segmentation variables. In 2009 Gastón Sánchez proposed an algorithm PATHMOX with the aim to automatic detecting heterogeneous segments within the PLS-PM methodology. This technique, based on recursive partitioning, produces a segmentation tree with a distinct path models in each node. At each node PATHMOX searches among all splits based on the segmentation variables and chooses the one resulting in the maximal difference between the PLS-PM models in the children nodes. Starting from the work of Sanchez the purpose of the thesis is to extend PATHMOX in the following points:
1. Extension to the PATHMOX approach to detect which constructs differentiate segments. The PATHMOX approach uses a F-global test to identify the best split in heterogeneous segments. Following the same approach it is possible to extend the testing to find which the endogenous constructs are and which are the relationships between constructs responsible of the difference between the segments.
2. Extension to the PATHMOX approach to deal with the factor invariance problem. Originally PATHMOX adapted the estimation of constructs to each detected segment, that is, once a split is performed the PLS-PM model is recalculated in every child. This leads to the problem of invariance: if the the estimation of the latent variables are recalculated in each terminal node of the tree, we cannot be sure to compare the distinct behavior of two individuals who belong to two different terminal nodes. To solve this problem we will propose a invariance test based on the X^2 distribution, where the goal of to test whether the measurement models of each terminal node can be considered equal or not among them.
3. Extension to the PATHMOX approach to overcome the parametric hypothesis of F-test. One critic to the PATHMOX approach, applied in the context of partial least square path modeling, is that it utilizes a parametric test based on the hypothesis that the residuals have a normal distribution to compare two structural models. PLS-PM in general, is utilized to model data that come from survey analysis. These data are characterized by an asymmetric distribution. This situation produces skewness in the distribution of data. As we know, PLS-PM methodology, is based in the absence of assumptions about the distribution of data. Hence, the parametric F test used in PATHMOX may represent a limit of the methodology. To overcome this limit, we will extend the test in the context of LAD robust regression.
4. Generalization of PATHMOX algorithm to any type of modeling methodology. The PATHMOX algorithm has been proposed to analyze heterogeneity in the context of the partial least square path modeling. However, this algorithm can be applied to many other kind of methodologies according to the appropriate split criterion. To generalize PATHMOX we will consider three distinct scenarios: Regression analysis (OLS, LAD, GLM regression) and Principal Component Analysis.
5. Implement the methodology, using the R software as specific library.